The separability problem is one of the basic and emergent problems in present and future quantum information processing. The latter focuses on information and computing based on quantum mechanics and uses quantum bits as its basic information units. In this paper we present an overview of the progress in the separability problem in bipartite systems, more specifically in two quantum bits (qubits) system, from the criterion based on the Bell’s inequalities in 1964 to the Li-Qiao Criterion and the enhanced entanglement criterion based on the symmetric informationally complete positive operator valued measures (SIC POVMs) in 2018.

Multigranulation is a new approach to the Rough Set Theory, where several separability relationships are used to obtain different granulations of the universe. The Multigranulation starts from the existence of different contexts or subsets of features to characterize the objects of the universe. In this paper, a method for the generation of contexts from the construction of similarity relations is proposed.The proposed solution was evaluated in an international database using the KNN classifier. It was also applied in the solution of a real problem in Civil Engineering specifically in Traffic Engineering, the contexts generated from the proposal used to determine the features of higher incidence in the service level of the road. The results achieved both in the international database and in the proposed application demonstrate the applicability of the proposed method.

This work is driven by a practical question, corrections of Artificial Intelligence (AI) errors. Systematic re-training of a large AI system is hardly possible. To solve this problem, special external devices, correctors, are developed. They should provide quick and non-iterative system fix without modification of a legacy AI system. A common universal part of the AI corrector is a classifier that should separate undesired and erroneous behavior from normal operation. Training of such classifiers is a grand challenge at the heart of the one- and few-shot learning methods. Effectiveness of one- and few-short methods is based on either significant dimensionality reductions or the blessing of dimensionality effects. Stochastic separability is a blessing of dimensionality phenomenon that allows one-and few-shot error correction: in high-dimensional datasets under broad assumptions each point can be separated from the rest of the set by simple and robust linear discriminant. The hierarchical structure of data universe is introduced where each data cluster has a granular internal structure, etc. New stochastic separation theorems for the data distributions with fine-grained structure are formulated and proved. Separation theorems in infinite-dimensional limits are proven under assumptions of compact embedding of patterns into data space. New multi-correctors of AI systems are presented and illustrated with examples of predicting errors and learning new classes of objects by a deep convolutional neural network.

Quantum nonlocality represents correlation properties between subsystems of a composite quantum system, usually including the four types: Bell nonlocality, steerability, entanglement, and quantum correlation (quantum discord). Given a basis $e_{AB}=\{|e_{ij}\>\}_{i\in[d_A],j\in[d_B]}$ for the Hilbert space $\H_A\otimes \H_B$ of a bipartite system $AB$, a density operator $\rho^{AB}$ (quantum state) of $AB$ can be represented as a $d_Ad_B\times d_Ad_B$ matrix $\hat{\rho}_{e_{AB}}=[\]$, called the density matrix of a density operator $\rho^{AB}$. A natural question is what is the relationship between the quantum nonlocality of a density operator $\rho^{AB}$ and its corresponding density matrix $\hat{\rho}_{e_{AB}}$? In this work, we discuss the relationships between quantum locality and basis, and prove that one type of quantum locality of density operators and that of their density matrices under a basis are the same if and only if the chosen basis is the tensor product of the bases of subsystems. Consequently, different bases define different quantum nonlocality density operators.

In this paper, we study a differential operator of parabolic type with a variable and unbounded coefficient, defined on an infinite strip. Sufficient conditions for the existence and compactness of the resolvent are established, and an estimate for the maximum regularity of solutions of the equation Lu=f∈L_2(Ω) is obtained. Two-sided estimates for the distribution function of approximation numbers are obtained. As is known, estimates of approximation numbers show the rate of best approximation of the resolvent of an operator by finite-dimensional operators. The paper proves the assertion about the existence of positive eigenvalues among the eigenvalues of the given operator and finds two-sided estimates for them.

In this paper, the connections between quantum non-locality and permutation symmetries are explored. This includes two types of symmetries: permutation across a superposition and permutation of qubits in a quantum system. An algorithm is proposed for nding the separability class of a quantum state using a method based on factorizing an arbitrary multipartite state into possible partitions, cyclically permuting qubits of the vectors in a superposition to check which separability class it falls into and thereafter using a reduced density-matrix analysis of the system is proposed. For the case of mixed quantum states, conditions for separability are found in terms of the partial transposition of the density matrices of the quantum system. One of these conditions turns out to be the Partial Positive Transpose (PPT) condition. A graphical method for analyzing separability is also proposed. The concept of permutation of qubits is shown to be useful in dening a new entanglement measure in the `engle'.